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VVR 120 Fluid Mechanics
1. Introduction, fluid properties (1.1, and handouts)
• Introduction, general information
• Course overview
• Fluids as a continuum
• Density
• Compressibility
• Viscosity
Exercises: A1
VVR 120 Fluid Mechanics
Applications of fluid mechanics
• Societal supply of safe energy and water by gas and fluids in pipes and channels
• Energy production (oil, hydropower, nuclear energy, natural gas)
• Environmental engineering and water treatment (channels, basins, filtering)
• Industrial process technology (relationship temperature, pressure, and energy)
• Protection against climate extremes/catastrophes (flooding, harbours, wind forces)
• Biomedical engineering
• Ecological evolution of species (predator-prey)
VVR 120 Fluid Mechanics
Major ”Fluid mechanics” employers
• SWECO
• Thyréns
• WSPgroup
• Skanska
• Eon, VA SYD
• Community offices
• Governmental, Naturvårdsverket
VVR 120 Fluid Mechanics
Fluid mechanics
• Fluid properties (2)
• Hydrostatics (3)
• Basic equations (6)
• Pipe flow (5)
• Flow around submerged
bodies (1)
• Channel flow (3)
• Repetition (2)
VVR 120 Fluid Mechanics
FLUID AS A CONTINUUM
• A fluid is considered to be a continuum in which there
are no holes or voids velocity, pressure, and
temperature fields are continuous.
• Validity criteria: Smallest length scale in a flow >>
average spacing between molecules composing the
fluid.
VVR 120 Fluid Mechanics
VVR 120 Fluid Mechanics
DENSITY ()
Mass/ unit volume (kg/m3)
Density decreases normally with increasing temperaturewater = (T,S,p)
i.e., dependent on
- Temperature
- Salt content ( 1000 + 0.741S, S in per mille;
S = 3.5% in ocean = 1026 kg/m3)
- Pressure (but only a small variability)
VVR 120 Fluid Mechanics
OTHER DEFINITIONS• Weight = mass gravity acceleration
(W = mg, [N = kgm/s2]) (Eqn. 1.4)
• Weight density (or specific weight)= density gravity acceleration(w = g, [N/m3 = kg/(m2s2)]) (Eqn. 1.6) (Note w = γ in exercises)
• Specific volume = reciprocal of density( = 1/, [m3/kg])
• Relative density (or specific gravity), s, is the density normalized with the density of water at a specific temperature and pressure (normally 4C and atmospheric pressure):s = R.d. = /water (often = /1000) (Eqn. 1.7)
• Power P [W = J/s = kgm2/s3 = Nm/s]; P = T ω (T = torque, ω = angular velocity [rad/s, 360o = 2rad]; V = ω r (V = velocity, r = radius)
VVR 120 Fluid Mechanics
Example – density. The specific weight of
water at ordinary temperature and pressure is
9.81 kN/m3. The specific gravity of mercury is
13.56. Compute the density of water and the
specific weight and density of mercury.
VVR 120 Fluid Mechanics
COMPRESSIBILITY
• All fluids can be compressed by application of pressure
elastic energy being stored
• Modulus of elasticity describes the compressibility
properties of the fluid and is defined on the basis of
volume
VVR 120 Fluid Mechanics
• Modulus of elasticity:
E=-dp/(dV/V1) [Pa]
• For liquids, region of engineering interest is when V/V1 1
• Ewater ~ 2109 Pa (function of temperature)
E
p
V
V
VVR 120 Fluid Mechanics
A1 What pressure must be applied to
water to reduce its volume 1 % ?
VVR 120 Fluid Mechanics
Example – compressibility. At a depth of 8 km in
the ocean the pressure is 81.8 MPa. Assume that
the specific weight of sea water at the surface is
10.05 kN/m3 and that the average volume
modulus of elasticity is 2.34*109 N/m2 for the
pressure range.
A) What will be the change in specific volume
between that at the surface and at that depth?
B) What will be the specific volume at that depth?
C) What will be the specific weight at that depth
VVR 120 Fluid Mechanics
IDEAL FLUID
A fluid in which there is no friction
REAL FLUID
A fluid in which shearing forces always exist
whenever motion takes place due to the fluid’s
inner friction – viscosity.
VVR 120 Fluid Mechanics
VISCOSITY
• Viscosity is a measure of a fluid’s “inner friction” or
resistance to shear stress.
• It arises from the interaction and cohesion of fluid
molecules.
• All fluids posses viscosity, but to a varying degree. For
instance, syrup has a considerably higher viscosity than
water.
VVR 120 Fluid Mechanics
DEFINITION OF DYNAMIC VISCOSITY -
Shearing of thin fluid film between two plates. The upper plate has
an area A.
• Experiments have shown that for a large number of fluids:
F ~ AV/h (if V and h not too large)
• Linear velocity profile V/h = dv/dy
y
VVR 120 Fluid Mechanics
• Introduction of the proportionality constant , named
dynamic viscosity, gives Newton’s viscosity law shear
force:
[Pas or kg/ms]
• = / [m2/s] - Kinematic viscosity
• No-slip condition – water particles adjacent to solid
boundary has zero velocity (observational fact)
dy
dv
h
V
A
F
(Eqn. 4.1-4.2) N/m2
VVR 120 Fluid Mechanics
μ (Pa·s)
VVR 120 Fluid Mechanics
Implication of viscosity: a fluid cannot sustain a shear
stress without deformation
VVR 120 Fluid Mechanics
Implications of Newton’s law:
• , independent of pressure (in contrast to solids)
• no velocity gradient no shear stress
Restriction of Newton’s law:
• law only valid if the fluid flow is laminar in which viscous
action is strong
VVR 120 Fluid Mechanics
• Laminar flow: smooth, orderly motion in which fluid
elements appears to slide over each other in layers (little
exchange between layers).
• Turbulent flow: random or chaotic motion of individual
fluid particles, and rapid mixing and exchange of these
particles through the flow
Turbulent flow is most common in nature.
VVR 120 Fluid Mechanics
Newtonian – non-Newtonian fluidsExamples non-Newtonian fluids:
Plastics, blood, suspensions, paints, foods
ii dy
du ,
Shear vs. rate of strain re-
lations for non-Newtonian
fluids:
Bingham plastic
n>1: Shear-thickening fluid,
n<1: Shear-thinning fluid
n
dy
du)(
VVR 120 Fluid Mechanics
Example – density. The specific weight of water at ordinary temperature and pressure is 9.81 kN/m3. The
specific gravity of mercury is 13.56. Compute the density of water and the specific weight and density of
mercury.
VVR 120 Fluid Mechanics
A1 What pressure must be applied to water to reduce its volume 1 % ?
VVR 120 Fluid Mechanics
Example – compressibility. At a depth of 8 km in the ocean the pressure is 81.8 MPa.
Assume that the specific weight of sea water at the surface is 10.05 kN/m3 and that
the average volume modulus of elasticity is 2.34*109 N/m2 for the pressure range.
A) What will be the change in specific volume between that at the surface and at that
depth?
B) What will be the specific volume at that depth?
C) What will be the specific weight at that depth